(iii) It suffices to show that each reduced word which is a product of length at most 4 in \(B\)  is not in the kernel of \(\phi _i\) , which is equivalent to the girth of \(\mbox{Cay}(G_i,B_i)\)  being greater than 4.
By [1]} the girth of \(\mbox{Cay}(G_i,B_i)\)  is at least \(\frac{2}{3}\log _q|G_i|\geqslant i\) , which completes the proof.
